All Questions
Tagged with reference-request nt.number-theory
1,408 questions
23
votes
2
answers
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views
Dirichlet and the prime number theorem
I browsed Dirichlets Werke today and was kind of surprised by two remarks that he made on p. 354 (Über die Bestimmung ...) and p. 372 (Sur l'usage ...). In the second paper, he claims (my ...
- 32.6k
23
votes
3
answers
1k
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References for $K_{4k}(\mathbb{Z})$
Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...
- 17.6k
23
votes
1
answer
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Chapters 1--4 of the Artin-Tate notes on Class Field Theory
Emil Artin and John Tate held a seminar on class field theory at Princeton University in 1951--1952. Their notes were published in 1967 by Benjamin (New York), but the first four chapters covering (...
- 18.7k
23
votes
1
answer
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A list of proofs of the Hasse–Minkowski theorem
I am currently doing a project in which I intend to include the most insightful possible proof of the Hasse–Minkowski theorem (also known as the Hasse principle for quadratic forms, among other names) ...
22
votes
3
answers
2k
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Hecke equidistribution
For a prime $p\equiv 1\pmod{4}$, we can write $p=a^2+b^2=N(a+bi)$. Therefore
$$
a+bi=p^{1/2}e^{i\varphi}
$$
where $\varphi\in [0,2\pi]$. I know that Hecke proved that $\varphi$ is equidistributed. I ...
- 2,508
22
votes
2
answers
3k
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$p$-adic Langlands correspondence
Basic question: Is it correct that the $p$-adic Langlands correspondence is known for $GL_2$ only over $Q_p$ but not other $p$-adic fields? If so, I would like to request some light to be shed on this ...
- 3,867
22
votes
3
answers
7k
views
A recommended roadmap to Fermat's Last Theorem
I was inspired to undertake math as a career after watching a documentary on the proof of Fermat's Last Theorem. As such it's been a small goal of mine to understand Wiles et al's proof.
In a ...
- 1,458
22
votes
1
answer
2k
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Reference request: The first cohomology of SL(2,Z) with coefficients in homogeneous polynomials
Let $H_k$ be the vector space of degree $k$ homogeneous polynomials in two variables.I'm looking for a reference for the fact that $H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus E_{...
- 4,898
22
votes
1
answer
770
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Writing $3p$ when $p \equiv 1 \pmod{3}$ as a sum of two rational cubes. Is this result new? And what about its converse?
I recently discovered a proof of the following. Let $p$ be a prime that's $1 \bmod {3}$. Suppose that $p$ is not represented by the principal quadratic form $(1,9,81)$ of discriminant $-243$ (The ...
- 5,422
21
votes
2
answers
1k
views
Most squares in the first half-interval
It is well known that if $p$ is an odd prime, exactly one half of the numbers $1, \dots, p-1$ are squares in $\mathbb{F}_p$. What is less obvious is that among these $(p-1)/2$ squares, at least one ...
- 14.7k
21
votes
5
answers
5k
views
What arrangement of unit cubes minimizes surface area?
For each of these two questions, one can assume that the arrangements are polycubes (for which a definition can be found in the excerpt-image below).
Question A. How does one arrange $n$ unit cubes ...
- 7,831
21
votes
3
answers
3k
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Twin Prime Conjecture Reference
I'm looking for a reference which has the first statement of the twin prime conjecture. According to wikipedia, nova, and several other quasi-reputable resources it is Euclid who first stated it, but ...
- 1,588
21
votes
2
answers
2k
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Applications of number theory in dynamical systems
I am looking for references (or ways to find references) on significant and/or recent applications of techniques in number theory to problems in the areas of dynamical systems and nonlinear dynamics.
...
- 760
21
votes
1
answer
771
views
Covering a set with geometric progressions
Consider the set $S_n=\{1,2,\cdots ,n\}$. What is the minimum number of distinct geometric progressions that cover $S_n$? Let us call this number $a_n$. I was wondering about this number after doing a ...
- 1,090
20
votes
2
answers
4k
views
information-theoretic derivation of the prime number theorem
Motivation:
While going through a couple interesting papers on the Physics of the Riemann Hypothesis [1] and the Minimum Description Length Principle [2], a derivation(not a proof) of the Prime Number ...
- 3,871
20
votes
4
answers
2k
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Does the set of happy numbers have a limiting density?
A positive integer $n$ is said to be happy if the sequence
$$n, s(n), s(s(n)), s(s(s(n))), \ldots$$
eventually reaches 1, where $s(n)$ denotes the sum of the squared digits of $n$.
For example, 7 is ...
- 856
20
votes
2
answers
2k
views
On a result attributed to W. Ljunggren and T. Nagell
I've read in a number of places that, building on previous work of T. Nagell, W. Ljunggren proved in 1 that the Diophantine equation
$$\frac{x^{n}-1}{x-1} = y^{2}$$
doesn't admit solutions in ...
- 8,842
20
votes
1
answer
2k
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When complex conjugation lies in the center of a Galois group
Let $K \subseteq \mathbb{C}$ be a number field (I'm fixing an embedding), and assume $K/\mathbb{Q}$ is Galois with Galois group $G$. Let $\tau \in G$ denote complex conjugation. This question ...
- 1,499
20
votes
6
answers
4k
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Erik Westzynthius's cool upper bound argument: update?
Version 2 of this writeup is
available, and includes a newer and simple upper bound thanks to
MathOverflow 88777 as
well as indirect references to future writeups. Details of further work
...
- 13k
20
votes
1
answer
786
views
Representation theory of reductive groups in characteristic $p$ as a limit of the theories in characteristic $0$
This question is out of plain curiosity. The first sentence of Deligne's
Les corps locaux de caractéristique $p$, limites de corps locaux de caractéristique $0$ (1984) reads (in rough translation) as ...
- 18.7k
20
votes
1
answer
1k
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Curves over number fields with everywhere good reduction
My question is the following:$\newcommand{\Q}{\Bbb Q}
\newcommand{\Z}{\Bbb Z}$
What is known about number fields $K$ fulfilling the condition
$C_{g,K}$ "there is a smooth projective curve of ...
- 1,742
19
votes
1
answer
3k
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Mazur secret Bourbaki report "Analyse p-adique"
Does anyone happen to know if a scan of Mazur's report exists, and, if so, where to find it? It appears in the references for Katz's "Higher congruences" and "Eisenstein measure" papers.
19
votes
5
answers
2k
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Sum of the reciprocals of radicals
Recall that the radical of an integer $n$ is defined to be $\operatorname{rad}(n) = \prod_{p \mid n } p$.
For a paper, I need the result that
$$\sum_{n \leq x} \frac{1}{\operatorname{rad}(n)} \ll_\...
- 21.3k
19
votes
2
answers
2k
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Who first proved the generalization of Bertrand's postulate to (2n,3n) and (3n,4n)?
In Wikipedia's page for Bertrand's postulate, it is said that its (2n,3n) version was proved by El Bachraoui in 2006. Seems likely that it was first proved way before than that! Can anyone point to ...
- 2,992
19
votes
1
answer
1k
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Deligne's letter to Bhargava from March 2004
I am quite interested in moduli spaces for Rings and Ideals, a letter from Deligne to Bhargava is cited in Melanie Wood's thesis Moduli spaces for Rings and Ideals (pdf), studying the minimal free ...
- 461
19
votes
3
answers
1k
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Points of elliptic curves over cyclotomic extensions
Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It ...
- 2,305
19
votes
3
answers
2k
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Cyclotomic polynomials: $\Phi_n(p)$ is like $p^{\phi(n)}$ for big enough $p$, right?
Apologies in advance if this turns out to be simple. So far I haven't found a proof or a reference.
Although I like $p$ to be a prime, I can ask the following for positive integers $n$ and $p$, ...
- 3,209
19
votes
1
answer
2k
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Has anything ever been done with the set $\{1,2,3,4,\ldots\}$ equipped with the operation $a \oplus b = a+b-1$ and the usual notion of multiplication?
Definition. $$\mathbb{J} = \{1,2,3,\ldots\}.$$
We can refer to the elements of $\mathbb{J}$ as "joiners."
The product of joiners is inherited from $\mathbb{Z}$.
The sum of joiners will be ...
- 3,793
19
votes
1
answer
2k
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"The Galois group of $\pi$ is $\mathbb{Z}$."
Last year, in a talk of Michel Waldschmidt's, I remember hearing a statement along the lines of the title of this question, that is, "The Galois group of $\pi$ is $\mathbb{Z}$.". In what sense/...
- 1,105
19
votes
1
answer
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Definitions of $\pi_1 \times \pi_2, \pi_1 \boxplus \pi_2, \pi_1 \boxtimes \pi_2$
Let $\pi_i$ be a smooth, admissible (possibly irreducible) representation of $\operatorname{GL}_{n_i}(k)$ for $k$ a $p$-adic field. I have seen the following representations defined in terms of $\...
- 6,180
19
votes
1
answer
2k
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Legendre and sums of three squares
The Three-Squares-Theorem was proved by Gauss in his Disquisitiones, and this proof was studied carefully by various number theorists. Three years before Gauss, Legendre claimed
to have given a proof ...
- 32.6k
19
votes
1
answer
1k
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Ehresmann's theorem over the $p$-adics
I am looking for a version of Ehresmann's theorem for analytic manifolds over the $p$-adic numbers $\mathbb{Q}_p$ or, more generally, local fields. I follow the conventions from Serre's book "Lie ...
- 21.3k
19
votes
0
answers
523
views
univariate integer version of Hilbert's 17th problem
Let $f(x)$ be a polynomial of degree $d$ with integer coefficients such that $f(x)\geqslant 0$ for all real $x$. Is it necessarily true that there exists an integer $N(d)$ such that $N(d)\cdot f$ is a ...
- 109k
18
votes
2
answers
3k
views
Only odd primes?
For $k \ge 2$, let
$$u = \{\lfloor{(k - \sqrt{k})n}\rfloor : n \ge 1\}$$
$$v = \{\lfloor{(k + \sqrt{k})n}\rfloor : n \ge 1\}.$$
My computer suggests that $u$ and $v$ are disjoint if and only if $k$ is ...
- 479
18
votes
6
answers
2k
views
Explicit formula for the trace of an unramified principal series representation of $GL(n,K)$, $K$ $p$-adic.
Let $K$ be a non-arch local field (I'm only interested in the char 0 case), let $\mathbb{G}$ be a connected reductive group over $K$ and let $G=\mathbb{G}(K)$. If $V$ is a smooth irreducible complex ...
- 41.4k
18
votes
5
answers
3k
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An elementary, short proof that the group of units of the ring of integers of a number field is finitely generated
Dirichlet's unit theorem states that (i) the group of units, $\mathscr{U}_K$, of the ring of integers of a number field $K$ is finitely generated, and (ii) the rank of $\mathscr{U}_K$ is equal to $r_1 ...
- 10.5k
18
votes
3
answers
6k
views
The multiplicative order of 2 modulo primes
Artin's Conjecture says that any positive integer, which is not a square, is a primitive root modulo infinitely many primes. Christopher Hooley gave in
Hooley, Christopher (1967). "On Artin's ...
- 25.5k
18
votes
2
answers
3k
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References for Artin motives
I find the following description of Artin motives in Wikipedia. Since these seem to be quite related to number theory, I am interested to learn more in that context. I request the experts available in ...
- 7,442
18
votes
1
answer
1k
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Distinct integer roots for a degree 7+ polynomial and its derivative
Question: Is there a polynomial $f \in \mathbb{Z}[x]$ with $\deg(f) \geq 7$ such that
all roots of $f$ are distinct integers; and
all roots of $f'$ are distinct integers?
Background:
I asked a ...
- 7,831
18
votes
3
answers
745
views
Number of primitive $n$th roots with positive versus negative real parts
Does anyone know a reference to the following results, which I can prove, but I suspect may be known. Let $R(n)$ denote the number of primitive $n$th roots of unity with positive real part, and $L(n)$ ...
- 1,991
18
votes
1
answer
631
views
Best texts on Lie groups for number theorists
What are the most comprehensive textbooks on the structure of Lie groups and their infinite-dimensional representations if one is interested in their applications to number theory (so covering ...
- 221
18
votes
1
answer
562
views
Is special value of Epstein zeta function in 3 variables a period?
Kontsevich-Zagier's article "Periods" contains the following question
Is $\displaystyle \sum_{x,y,z \in \mathbb{Z}}' \frac{1}{(x^2+y^2+z^2)^2}$ an extended period?
($\sum'$ means we do not sum ...
- 809
18
votes
0
answers
718
views
Erdos-Kac for squarefree numbers
In its usual form, the Erdos-Kac Theorem states that if $f(n) : \mathbb{N} \rightarrow \mathbb{R}$ is a strongly additive function with $|f(p)| \le 1$ for all primes $p$, then
$$\frac{|\{n \le x : \...
- 211
17
votes
13
answers
6k
views
Probability in number theory
I am hearing that there are some great applications of probability theory (or more general measure theory) to number theory. Could anyone recommend some good book(s) on that (or other types of ...
17
votes
5
answers
4k
views
Fermat numbers and the infinitude of primes
Wonder whether any of you guys know why it is that the proof of the infinitude of primes that is based on the coprimality of any pair of (distinct) Fermat numbers is commonly attributed to Pólya.
In ...
- 8,842
17
votes
2
answers
3k
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Consequences of the Birch and Swinnerton-Dyer Conjecture?
Before asking my short question I had made some research. Unfortunately I did not find a good reference with some examples. My question is the following
What are the consequences of the Birch and ...
17
votes
1
answer
3k
views
Is $x^{n}-x-1$ irreducible?
Is it true that for every $n \in \mathbb{N}$, $x^{n}-x-1$ is irreducible in $\mathbb{Z}[x]$?
The standard irreducibility criteria seem to fail.
- 11.3k
17
votes
1
answer
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Is there an analog of the Birch/Swinnerton-Dyer conjecture for abelian varieties in higher dimensions?
I am wondering if there is a multi-dimensional analog of the
Birch/Swinnerton-Dyer (BSD) conjecture.
The recent famous result inching toward resolution of that conjecture is:
Bhargava, Manjul, and ...
- 151k
17
votes
2
answers
1k
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The GCD-matrix: generalizing a result of Smith?
Let $M$ be the $n\times n$ matrix, known as the GCD matrix, of entries $M_{ij}=\gcd(i,j)$. In the paper
H J S Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7:208-...
- 43.2k
17
votes
2
answers
2k
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Why does Tate's conjecture imply semisimplicity of crystalline Frobenius?
I'm trying to find a reference for the following fact:
If Tate's conjecture is true for all smooth projective varieties over $\mathbb{F}_p$, then the Frobenius endomorphism on the crystalline ...
- 37k